# Poisson: Create a Poisson distribution In distributions3: Probability Distributions as S3 Objects

 Poisson R Documentation

## Create a Poisson distribution

### Description

Poisson distributions are frequently used to model counts.

### Usage

```Poisson(lambda)
```

### Arguments

 `lambda` The shape parameter, which is also the mean and the variance of the distribution. Can be any positive number.

### Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Poisson random variable with parameter `lambda` = λ.

Support: {0, 1, 2, 3, ...}

Mean: λ

Variance: λ

Probability mass function (p.m.f):

P(X = k) = λ^k e^(-λ) / k!

Cumulative distribution function (c.d.f):

P(X ≤ k) = e^(-λ) ∑_{i = 0}^k λ^i / i!

Moment generating function (m.g.f):

E(e^(tX)) = e^(λ (e^t - 1))

### Value

A `Poisson` object.

Other discrete distributions: `Bernoulli()`, `Binomial()`, `Categorical()`, `Geometric()`, `HurdleNegativeBinomial()`, `HurdlePoisson()`, `HyperGeometric()`, `Multinomial()`, `NegativeBinomial()`, `ZINegativeBinomial()`, `ZIPoisson()`, `ZTNegativeBinomial()`, `ZTPoisson()`

### Examples

```
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
```

distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.